Move object along multiple paths
Conventionally, (a bit like grabbing/releasing weapons, etc, if you want to look up other examples,) set up two or more Empties with the desired motion and timings for the car. Each empty can have its own Follow Path constraint. Ensure they are located and rotated similarly at hand-over points.
Give the car a Child Of constraint for each Empty, and for an instantaneous hand-over from one Empty to another, keyframe the Influences of the appropriate Child Of constraints to 0 and 1 respectively, in consecutive frames.
(Use the Set Inverse switches to ensure the car is all lined up, ready for hand-over)

If you need further control over rotations, moves, you can add another layer of indirection / hierarchy between the constrained Empties and the car: then the car would be constrained to be a grandchild of the path-following Empties.